Affine Difference Equations and Long-Term Behavior
“Cobweb” diagrams
Affine Difference
Equations---Slope
bigger than 1
What if we start iterating
with a point that lies to the
left
of the fixed point?
Affine Difference
Equations---Slope
smaller than 1
Affine Difference Equations---Slope equal to1
Conclusions:
Long term behavior of solutions to affine difference equations:
If , the sequence (
A
(
n
)) ,
n
= 1, 2, 3,. . . “blows
up”. That is,
The fixed point is a
repelling
fixed point.
If , the sequence (
A
(
n
)) ,
n
= 1, 2, 3,. . .
Converges to the fixed point of the function. That is,
The fixed point is an
attracting
fixed point.
Exploring different slopes in affine difference equations and their impact on the behavior of solutions. Discover how iterating with points relative to fixed points reveals attracting or repelling characteristics. Gain insights into the convergence or divergence of sequences in relation to fixed points in the functions.
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Presentation Transcript
= ( ) f x y Affine Difference Equations---Slope bigger than 1 = y x n a
= ( ) f x y Affine Difference Equations---Slope bigger than 1 = y x n a
Affine Difference Equations---Slope less than -1 = y x n a = ( ) f x y
Affine Difference Equations---Slope smaller than 1 = y x = ( ) f x y n a fixed pt What if we start iterating with a point that lies to the left of the fixed point?
Affine Difference Equations---Slope in (-1,0). = y x n a fixed pt = ( ) f x y
Affine Difference Equations---Slope equal to1 = ( ) f x x + = ( 1) ( ) A n A n = + ( ) f x x k + = + ( 1) ( ) A n A n k
Conclusions: Long term behavior of solutions to affine difference equations: A( 1) ( ) iterating n kA n b + = + = kx b + ( ) f x | | 1 k If , the sequence (A(n)) , n = 1, 2, 3,. . . blows up . That is, The fixed point is a repelling fixed point. If , the sequence (A(n)) , n = 1, 2, 3,. . . Converges to the fixed point of the function. That is, ( ) A n as n 0 | | 1 k b ( ) A n as n 1 k The fixed point is an attracting fixed point.
